Implicit-explicit (IMEX) methods for the incompressible flows with variable viscosity
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In implicit-explicit (IMEX) temporal discretisations of the Navier--Stokes equations, the viscous term is usually kept implicit, while convection is treated either explicitly or semi-implicitly. Such methods are very popular, as they simplify implementation, improve computational efficiency and retain good numerical stability. For flows with non-constant viscosity, as arising in various practical applications, it can be attractive to treat also the viscous term in a semi-implicit manner. This allows us, for instance, to use simpler solvers by avoiding additional terms that would otherwise couple the velocity components. In this talk, we present different IMEX treatments of the viscous term and discuss the implications of such approaches. Our methods are optimally convergent and unconditionally stable, that is, the explicitisation does not incur any CFL-like conditions. Both monolithic and fractional-step methods are considered, with numerical examples corroborating our theory.
